Suppose a random variable X such that E (X2) = 5E (X)-6. Find the interval in...
Suppose X is a random variable such that E(X) and E(X2 ) both exist, and are finite. Consider the function f(c) of a real number c given by f(c) = E[(X ? c)2 ]. (a) (2 pts.) Find this function f(c) when X ? Bin(3, 1/2). Among the ’zoo’ of functions that you know about, what kind of function is it? (b) (8 pts.) Find the value of c which MINIMIZES the function f(c). Hint: expand out the (X ?...
10. Suppose that a random variable X has the uniform distribution on the interval [-2,8). Find the pdf of X and the value of P(O<X<7).
6. Suppose that Xi, X2. Xn are independent random variable thal are uniformly distributed in the unit interval (0, Let Y maxXi, X2Xnbe their maximum value. Determine the disiribution function and the density of Y and thence evaluale E(Y) and Var(Y
15. Let the random variables X1 and X2 be the payoffs of two different Suppose E(X;) = E(X2) = 100, and V( X) = V(X;) = investments 10. Suppose an investor owns 50% of each investment so the total payoff is: (X1+ X2 ) /2. There is a fixed fee (brokerage fee, for example) of 15 to acquire the two investments. So the investor's net payoff is: (Xi+X2)/2 - 15 a) Is the investor's net payoff a random variable? If...
• Suppose X, and X2 are Independent random Variables with Exi) = E(X2) - 1, V(X)=1 and V(X2) 24. ca Find v(2X-X2). (b) Find CoV (X,+ X2 +2 , X, X2 ).. (c) Findcov (x1+x2 + 2 X, - X2+3). do Find v(x,x2). (e) Find cov (X1, X, X2).
Suppose X is a random variable that has density function f(x) = (1/2)e^−|x| for −∞ < x < ∞. Find: (a) (2 pts) P(X < 10). (b) (4 pts) The c.d.f. of X2. (c) (4 pts) V ar(X)
Suppose X is a continuous random variable having pdf (1+x, -1 < x < 0, f(x) = { 1 – x, 0 < x <1, lo, otherwise (a) Find E(X2). (b) Find Var(X2).
Suppose we assume that X1, X2, . . . , Xn is a random sample from a「(1, θ) distribution a) Show that the random variable (2/0) X has a x2 distribution with 2n degrees of freedom. (b) Using the random variable in part (a) as a pivot random variable, find a (1-a) 100% confidence interval for
2. Let Xi, X2,...,Xn be independent, uniformly distributed random variables on the interval 0,e (a) Find the pdf of X(), the jth order statistic. b) Use the result from (a) to find E(X)). the mean difference between two successive order statistics (d) Suppose that n- 10, and X.. , Xio represent the waiting times that the n 10 people must wait at a bus stop for their bus to arrive. Interpret the result of (c) in the context of this...
6. [ 10 pts.] Let X be a random variable with and σ,= 4. Find the following quantities: a. E[2X - 4] b. E[X2] c. E[4X2]